{"title":"On points avoiding measures","authors":"Piotr Borodulin–Nadzieja , Artsiom Ranchynski","doi":"10.1016/j.topol.2024.108988","DOIUrl":"10.1016/j.topol.2024.108988","url":null,"abstract":"<div><p>We say that an element <em>x</em> of a topological space <em>X</em> avoids measures if for every Borel measure <em>μ</em> on <em>X</em> if <span><math><mi>μ</mi><mo>(</mo><mo>{</mo><mi>x</mi><mo>}</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, then there is an open <span><math><mi>U</mi><mo>∋</mo><mi>x</mi></math></span> such that <span><math><mi>μ</mi><mo>(</mo><mi>U</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. The negation of this property can viewed as a local version of the property of supporting a strictly positive measure. We study points avoiding measures in the general setting as well as in the context of <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the remainder of Stone-Čech compactification of <em>ω</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108988"},"PeriodicalIF":0.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141404552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some remarks on Erdős spaces","authors":"Alfredo Zaragoza","doi":"10.1016/j.topol.2024.108987","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108987","url":null,"abstract":"<div><p>The objective of this work is to present some results related to some Erőds spaces. This paper answers a question made by the author in <span>[12]</span> proving that if <em>X</em> is a cohesive space then <span><math><mi>K</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a cohesive space; we give a partial answer to question 7.3 of <span>[7]</span> providing an internal characterization of <span><math><mi>Q</mi><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>-factors for certain subsets of <span><math><mi>Q</mi><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>; and we give conditions under which a perfect or open image of the complete Erdős space is homeomorphic to the complete Erdős space.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108987"},"PeriodicalIF":0.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141323269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Connectedness of certain graph coloring complexes","authors":"Nandini Nilakantan , Samir Shukla","doi":"10.1016/j.topol.2024.108985","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108985","url":null,"abstract":"<div><p>In this article, we consider the bipartite graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We prove that the connectedness of the complex <span><math><mtext>Hom</mtext><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mi>m</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> if <span><math><mi>m</mi><mo>≥</mo><mi>n</mi></math></span> and <span><math><mi>m</mi><mo>−</mo><mn>3</mn></math></span> in all the other cases. Therefore, we show that for this class of graphs, <span><math><mtext>Hom</mtext><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> is exactly <span><math><mo>(</mo><mi>m</mi><mo>−</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>-connected, <span><math><mi>m</mi><mo>≥</mo><mi>n</mi></math></span>, where <em>d</em> is the maximal degree of the graph <em>G</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108985"},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Vector bundles over non-Hausdorff manifolds","authors":"David O'Connell","doi":"10.1016/j.topol.2024.108982","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108982","url":null,"abstract":"<div><p>In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of standard vector bundles. We then use this description to introduce various formulas that express non-Hausdorff structures in terms of data defined on certain Hausdorff submanifolds. Finally, we use Čech cohomology to classify the real non-Hausdorff line bundles.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108982"},"PeriodicalIF":0.6,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124001676/pdfft?md5=c27d5e13349e4e55067738836fba7460&pid=1-s2.0-S0166864124001676-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141323268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On idempotent convexities and idempotent barycenter maps","authors":"Dawid Krasiński , Taras Radul","doi":"10.1016/j.topol.2024.108974","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108974","url":null,"abstract":"<div><p>We consider an isomorphism between the idempotent convexity based on the maximum and the addition operations and the idempotent measure convexity on the maximum and the multiplication operations. We use this isomorphism to investigate topological properties of the barycenter map related to the maximum and the multiplication operations.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"353 ","pages":"Article 108974"},"PeriodicalIF":0.6,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the arc-wise connection relation in the plane","authors":"Gabriel Debs, Jean Saint Raymond","doi":"10.1016/j.topol.2024.108975","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108975","url":null,"abstract":"<div><p>We prove that the arc-wise connection relation in a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span> subset of the plane is Borel.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"353 ","pages":"Article 108975"},"PeriodicalIF":0.6,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141263956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Florencio Corona-Vázquez, José A. Martínez-Cortez, Russell-Aarón Quiñones-Estrella, Javier Sánchez-Martínez
{"title":"About the hyperspace H(X)/H(X;K)","authors":"Florencio Corona-Vázquez, José A. Martínez-Cortez, Russell-Aarón Quiñones-Estrella, Javier Sánchez-Martínez","doi":"10.1016/j.topol.2024.108972","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108972","url":null,"abstract":"<div><p>Let <em>X</em> be a continuum, <em>K</em> a nonempty closed subset of <em>X</em>, and let <em>n</em> be a positive integer. In this paper, we consider the hyperspaces <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>X</mi></mrow></msup></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, consisting of all nonempty closed subsets of <em>X</em> and of all nonempty closed subsets of <em>X</em> having at most <em>n</em> components, respectively. If <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><mo>{</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>X</mi></mrow></msup><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></math></span>, <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>;</mo><mi>K</mi><mo>)</mo></math></span> denotes the hyperspace of all elements in <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> intersecting <em>K</em>. In this paper we present some topological properties of the quotient space <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><mi>H</mi><mo>(</mo><mi>X</mi><mo>;</mo><mi>K</mi><mo>)</mo></math></span>, going forward in its study in the available literature. In the class of finite graphs, we study the problem of determining conditions on <em>X</em> and <em>K</em> such that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>;</mo><mi>K</mi><mo>)</mo></math></span> are homeomorphic, obtaining in this direction some characterizations.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"353 ","pages":"Article 108972"},"PeriodicalIF":0.6,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141263960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polygons inscribed in Jordan curves with prescribed edge ratios","authors":"Yaping Xu , Ze Zhou","doi":"10.1016/j.topol.2024.108971","DOIUrl":"10.1016/j.topol.2024.108971","url":null,"abstract":"<div><p>Let <em>J</em> be a simple closed curve in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> <span><math><mo>(</mo><mi>k</mi><mo>≥</mo><mn>2</mn><mo>)</mo></math></span> that is differentiable with non-zero derivative at a point <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>J</mi></math></span>. For a tuple of positive reals <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> <span><math><mo>(</mo><mi>n</mi><mo>≥</mo><mn>3</mn><mo>)</mo></math></span>, each of which is less than the sum of the others, we show that there exists a polygon <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> inscribed in <em>J</em> with sides of lengths proportional to <span><math><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>. As a consequence, we prove the existence of triangle inscribed in <em>J</em> similar to any given triangle.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108971"},"PeriodicalIF":0.6,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spaces determined by countably many locally compact subspaces","authors":"Jimmie Lawson , Xiaoquan Xu","doi":"10.1016/j.topol.2024.108973","DOIUrl":"10.1016/j.topol.2024.108973","url":null,"abstract":"<div><p>The theory of compactly generated spaces, alternatively <em>k</em>-spaces, plays an important role in general and algebraic topology. In this paper, we develop a theory of compact generation for non-Hausdorff spaces using locally compact spaces. We are particularly interested in the case that countably many locally compact spaces suffice and restrict our attention to that case. The notion of <span><math><mi>ℓ</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-spaces (which can be considered as a type of <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-spaces) is introduced, which are determined by countably many locally compact spaces. It is proved that the product space of two <span><math><mi>ℓ</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-spaces is still an <span><math><mi>ℓ</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-space. We slightly modify the notion of an <span><math><mi>ℓ</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-space to apply it to posets equipped with the Scott topology. The notions of <span><math><mi>ℓ</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-posets, <span><math><mi>ℓ</mi><msub><mrow><mi>f</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-posets and <em>c</em>-posets are introduced. We develop a corresponding theory for these three kinds of posets and investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Several such conditions are given.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108973"},"PeriodicalIF":0.6,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mosaics for immersed surface-links","authors":"Seonmi Choi , Jieon Kim","doi":"10.1016/j.topol.2024.108961","DOIUrl":"10.1016/j.topol.2024.108961","url":null,"abstract":"<div><p>The concept of a knot mosaic was introduced by Lomonaco and Kauffman as a means to construct a quantum knot system. The mosaic number of a given knot <em>K</em> is defined as the minimum integer <em>n</em> that allows the representation of <em>K</em> on an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> mosaic board. Building upon this, the first author and Nelson extended the knot mosaic system to encompass surface-links through the utilization of marked graph diagrams and established both lower and upper bounds for the mosaic number of the surface-links presented in Yoshikawa's table. In this paper, we establish a mosaic system for immersed surface-links by using singular marked graph diagrams. We also provide the definition and discussion on the mosaic number for immersed surface-links.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"353 ","pages":"Article 108961"},"PeriodicalIF":0.6,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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